Finite Geometries

Peter Dembowski was born in Berlin on April 1, 1928. After studying mathematics at the University of Frankfurt am Main, he pursued his graduate studies at Brown University and at the University of Illinois, mainly with Reinhold Baer. Dembowski returned to Frankfurt in 1956. Shortly before his premature death in January 1971, he had been appointed to a chair at the University of Tübingen. Dembowski taught at the universities of Frankfurt and Tübingen and - as visiting professor - in London (Queen Mary College), Rome, Madison, WI, and Chicago, IL. Dembowski's chief research interest lay in the connections between finite geometries and group theory. His book "Finite Geometries", brought together essentially all that was known at that time about finite geometrical structures, including key results of the author, in a unified and structured perspective. This book became a standard reference as soon as it appeared in 1968. It influenced the expansion of combinatorial geometric research, and also left its trace in neigh-bouring areas.

1. Basic concepts.- 1.1 Finite incidence structures.- 1.2 Incidence preserving maps.- 1.3 Incidence matrices.- 1.4 Geometry of finite vector spaces.- 2. Designs.- 2.1 Combinatorial properties.- 2.2 Embeddings and extensions.- 2.3 Automorphisms of designs.- 2.4 Construction of designs.- 3. Projective and affine planes.- 3.1 General results.- 3.2 Combinatorics of finite planes.- 3.3 Correlations and polarities.- 3.4 Projectivities.- 4. Collineations of finite planes.- 4.1 Fixed elements and orders.- 4.2 Collineation groups.- 4.3 Central collineations.- 4.4 Groups with few orbits.- 5. Construction of finite planes.- 5.1 Algebraic representations.- 5.2 Planes of type IV.- 5.3 Planes of type V.- 5.4 Planes of types I and II.- 6. Inversive planes.- 6.1 General definitions and results.- 6.2 Combinatorics of finite inversive planes.- 6.3 Automorphisms.- 6.4 The known finite models.- 7. Appendices.- 7.1 Association schemes and partial designs.- 7.2 Hjelmslev planes.- 7.3 Generalized polygons.-7.4 Finite semi-planes.- Dictionary.- Special notations.

"Such a vast amount of information as this book contains can only be accomplished in 375 pages by a very economical style of writing... it enables one to have a good look at the forest without being too detracted by the individual trees... The author deserves unstinting praise for the skill, energy, and perseverance which he devoted to this work. The finished product confirms what his many earlier contributions to the subject of finite geometry have already indicated, namely, that he is an undisputed leader in his field." -Mathematical Reviews

"Such a vast amount of information as this book contains can only be accomplished in 375 pages by a very economical style of writing... it enables one to have a good look at the forest without being too detracted by the individual trees... The author deserves unstinting praise for the skill, energy, and perseverance which he devoted to this work. The finished product confirms what his many earlier contributions to the subject of finite geometry have already indicated, namely, that he is an undisputed leader in his field." -Mathematical Reviews